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The Tidal Response of Rocky and Ice-Rich Exoplanets G Astronomy & Astrophysics manuscript no. Tobie-AA_Tides_exoplanets c ESO 2018 August 30, 2018 The tidal response of rocky and ice-rich exoplanets G. Tobie1, O. Grasset1, C. Dumoulin1, and A. Mocquet1 Laboratoire de Planétologie et Géodynamique, UMR-CNRS 6112, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France e-mail: [email protected] Submitted Sept. xx, 2018 ABSTRACT The amount of detected planets with size comparable to the Earth increases drastically. Most of the Earth-size planet candidates orbit at close distances from their central star, and therefore are subjected to large tidal forcing. Accurate determination of the tidal parameters of exoplanets taking into account their interior structure and rheology is essential to better constrain their rotational and orbital history. In the present study, we compute the tidal response of rocky and ice-rich solid exoplanets for masses ranging between 0.5 and 10 Earth masses using a multilayer approach and an Andrade rheology. We showed that the amplitude of tidal response, characterized by the gravitational Love number, k2, depends mostly on the planet mass and the internal stratification (mostly controlled by the bulk composition). We showed that, for a given planet composition, the tidal Love number, k2, is mostly controlled by self-gravity effect and increase as a function of planet mass. For rocky planets, k2 depends mostly on the relative size of the iron core, and hence on the bulk iron fraction. For ice-rich planets, the presence of outer ice layers reduces the amplitude of tidal response compared to ice-free rocky planets. For both types of planet (rocky and ice-rich), we propose relatively simple scaling laws to predict the Love number as a function of radius, planet mass and composition. For the dissipation rate, characterized by the Q factor, we did not find any direct control by the planet mass. The dissipation rate is mostly sensitive to the forcing frequency and the internal viscosity, which depends on the planet composition and size. Key words. Exoplanets – Interiors – Tides 1. Introduction in the TRAPPIST-1 system, even if the orbital eccentricity is estimated to be small, strong internal heating, exceeding The number of detected exoplanets with radius and mass com- radiogenic heating, can be generated for the closest planets (e.g. parable to the Earth is now increasing drastically (e.g. Bonfils Turbet et al. 2018). The impact of tidal interactions on planetary et al. 2013; Batalha et al. 2013; Marcy et al. 2014; Dressing & evolution depend obviously on the orbital configuration of the Charbonneau 2015; Coughlin et al. 2016; Dittmann et al. 2017; system, but also on the way planetary bodies deform under the Gillon et al. 2017). Most of the low-mass planet candidates action of tides raised by the central stars, and in some cases by orbit at relatively close distance from their central star and planetary companion, like in the Earth-moon system. therefore are likely subjected to large tidal forcing. Low mass planets (2 − 10ME) with short orbital periods (< 10 − 20 days) seem especially abundant around M-dwarf stars (e.g. Bonfils The response of planetary interiors to tidal forcing is et al. 2013; Dressing & Charbonneau 2015). For example, the dependent on the internal structure and on the mechanical TRAPPIST-1 system recently discovered by Gillon et al. (2016, properties of each layer composing the planetary interior. Many 2017) exhibits several small planets orbiting a low-mass star at models have been developed to compute the tidal response of relatively close distance (< 0:1 AU), corresponding to orbital a variety of planetary objects of the Solar system in the past periods of a few Earth days. using multi-layer methods (e.g. Alterman et al. 1959; Kaula 1964; Sabadini et al. 1982; Segatz et al. 1988; Tobie et al. 2005; Such multi-planet system evolves mainly due to the grav- Wahr et al. 2009; Beuthe 2013; Dumoulin et al. 2017). Most itational tide raised by the star in the planets (the planetary of studies dedicated to exoplanets used simplified approach to tide). The planetary tide mainly acts to decrease the obliquity predict the tidal response assuming, for instance, the formula of the planet, synchronize the rotation and on longer timescales derived for homogenous viscoelastic interiors (Henning et al. decrease the eccentricity and semi-major axis. Therefore, tidal 2009; Efroimsky 2012; Makarov & Efroimsky 2014; Barr et al. evolution has a strong impact on the climate stability of the plan- 2017), thus neglecting the effect of density stratification and the ets and their habitability (e.g. Lammer et al. 2009; Barnes 2017; mechanical coupling between the different internal layers. To Turbet et al. 2018). Tidal friction occurring in the interior of our knowledge, only the study of Henning & Hurford (2014) such planets during tidal despinning as well as once the planet is used a multi-layer method to predict the tidal response of a tidally locked on an eccentric orbit can, in some circumstances, variety of exoplanets. However, Henning & Hurford (2014) significantly contribute to the internal heat budget, possibly considered interior models with constant values of density, enhancing internal melting and volcanism (Behounkovᡠet al. rigidity and viscosity for each internal layer 2011; Henning & Hurford 2014; Barr et al. 2017). For instance Article number, page 1 of 10 A&A proofs: manuscript no. Tobie-AA_Tides_exoplanets With the growing number of detected exoplanets, a variety magma ocean or liquid water ocean). This particular case will of structural models have been developed this last decade to be addressed in a future study. The input parameter are Fe/Si, predict the internal structure of massive rocky planets and large Mg/Si, the Mg content of the silicate mantle, the amount of icy worlds (e.g. Valencia et al. 2006; Sotin et al. 2007; Seager H2O and the total mass of the planet. For simplicity, we impose et al. 2007; Grasset et al. 2009; Swift et al. 2012; Weiss & a Mg content in the silicate mantle equal to the Earth’s one Marcy 2014; Dorn et al. 2015). These studies showed that the (defined as the mole fraction Mg/(Mg+Fe): Mg#=0.9) for mass-radius relationship is sensitive to the planet composition, the two classes of interior models. The reference value for so that accurate determination of planet mass and radius com- Fe/Si and Mg/Si are fixed to the solar value, 0.977 and 1.072 bined with stellar elemental abundances may potentially provide respectively. For the rock planets, the Fe/Si ratio is varying constraints on the core size, mantle composition and the water between δFe = −50% and +50% relative to the reference value. fraction of the planet. Even if the uncertainties are still relatively For ice-rich planets, the Fe/Si ratio is fixed to the reference high on the mass determination of small-size planets (e.g. Weiss value and only the ice fraction (xice is varied between 0 and 50% . & Marcy 2014), we can already observe some compositional tendencies. Rogers (2015) noticed, for instance, that most of The interior structure is modeled using the approach of Sotin planets with radius larger than 1.6 R⊕ have density lower than et al. (2007). The same equations of state (EoS) and parameters terrestrial planets suggesting a significant fraction of volatile, in are used for all layers, except for the surface liquid water layer form of H/He atmosphere and/or water layers. which replaces by low-pressure ices. The density profile in the upper part of the silicate mantle (P < 25 GPa) as well as in In models studying tidal evolution of planetary rotations and the low-pressure ice layers (P < 2:2 GPa) is computed with orbits (e.g. Mardling 2007; Bolmont et al. 2014; Barnes 2017), a Birch-Murnaghan EoS. For simplicity, no phase transition the effect of tidal damping is classically parametrized using is considered in the upper rock mantle (P<25 GPa) and the prescribed values of tidal Love number, k2, and dissipation low-pressure ice layers. These two low-pressure layers are function, Q−1 or time lag ∆t. In reality, the tidal response described with a single set of parameters. For the liquid iron is expected to depend on the planet composition, structure, core, the lower part of the silicate mantle and the high-pressure thermal state as well as on the frequency of the tidal forcing. ice layer (Ice VII), a Mie-Grunëisen-Debye EoS is employed. This requires to take into account the specificity of the interior The iron-rich core is assumed to be entirely liquid, no inner models and orbital configuration to predict the appropriate tidal solid core is considered. In all layers, the temperature profile parameters. is assumed to follow an isentropic temperature profile. Figure 1 displays the Mass-Radius relationship computed for interior In the present study, we perform systematic computation models with various iron and ice contents and the comparison of the tidal response for planet masses ranging between 0.5 with scaling laws using the polynomial coefficients provided in and 10 M⊕, having various Fe/Si ratio and H2O content. For Table 1. that purpose, we use a numerical code initially developed for computing the tidal response of icy moons (Tobie et al. 2005), recently applied to Venus and Earth’s application (Dumoulin et al. 2017). To determine the internal structure of planet as a function of mass and composition, we follow the method of Sotin et al. (2007) (see section 2). The rheological properties are adapted from the Earth’s case assuming an Andrade viscoelastic rheology and are extrapolated to high-pressure ranges (see section 2).
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